Chapter 10 Quadrilaterals

40 Pages·2007·0.34 MB·English

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CHAPTER
10
QUADRILATERALS
Euclid’s fifth postulate was often considered to be
a “flaw” in his development of geometry. Girolamo
Saccheri (1667–1733) was convinced that by the appli-
cation of rigorous logical reasoning, this postulate CHAPTER
could be proved. He proceeded to develop a geome- TABLE OF CONTENTS
try based on an isosceles quadrilateral with two base
10-1 The General Quadrilateral
angles that are right angles.This isosceles quadrilateral
10-2 The Parallelogram
had been proposed by the Persian mathematician
10-3 Proving That a Quadrilateral
Nasir al-Din al-Tusi (1201–1274).Using this quadrilateral,
Is a Parallelogram
Saccheri attempted to prove Euclid’s fifth postulate by
10-4 The Rectangle
reasoning to a contradiction. After his death,his work
10-5 The Rhombus
was published under the title Euclid Freed of Every Flaw.
10-6 The Square
Saccheri did not,as he set out to do,prove the paral-
lel postulate,but his work laid the foundations for new 10-7 The Trapezoid
geometries. János Bolyai (1802–1860) and Nicolai 10-8 Areas of Polygons
Lobachevsky (1793–1856) developed a geometry that Chapter Summary
allowed two lines parallel to the given line, through a
Vocabulary
point not on a given line.Georg Riemann (1826–1866)
Review Exercises
developed a geometry in which there is no line paral-
Cumulative Review
lel to a given line through a point not on the given line.
379
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380 Quadrilaterals
10-1 THE GENERAL QUADRILATERAL
Patchwork is an authentic American craft,developed by our frugal ancestors in
a time when nothing was wasted and useful parts of discarded clothing were
stitched into warm and decorative quilts.
Quilt patterns, many of which acquired
names as they were handed down from one
generation to the next,were the product of cre-
ative and industrious people. In the Lone Star
pattern, quadrilaterals are arranged to form
larger quadrilaterals that form a star.The cre-
ators of this pattern were perhaps more aware
of the pleasing effect of the design than of the
mathematical relationships that were the basis
of the pattern.
A quadrilateralis a polygon with four sides. R
In this chapter we will study the various special
Q
quadrilaterals and the properties of each.Let us
first name the general parts and state properties
of any quadrilateral,using PQRS as an example.
• Consecutive verticesor adjacent vertices P S
are vertices that are endpoints of the same
side such as P andQ,Q andR,R andS,S andP.
• Consecutive sidesor adjacent sidesare sides that have a common end-
point,such as PQand QR,QRand RS,RSand SP,SPand PQ.
• Opposite sides of a quadrilateralare sides that do not have a common
endpoint,such as PQand RS,SPand QR.
• Consecutive angles of a quadrilateralare angles whose vertices are consec-
utive,such as (cid:2)P and (cid:2)Q,(cid:2)Qand (cid:2)R,(cid:2)R and (cid:2)S,(cid:2)Sand (cid:2)P.
• Opposite angles of a quadrilateralare angles whose vertices are not con-
secutive,such as (cid:2)Pand (cid:2)R,(cid:2)Qand (cid:2)S.
• A diagonal of a quadrilateralis a line segment whose endpoints are two
nonadjacent vertices of the quadrilateral,such as PRand QS.
• The sum of the measures of the angles of a quadrilateral is 360 degrees.
Therefore,m(cid:2)P(cid:2)m(cid:2)Q(cid:2)m(cid:2)R(cid:2)m(cid:2)S (cid:3)360.
10-2 THE PARALLELOGRAM
DEFINITION
A parallelogram is a quadrilateral in which two pairs of opposite sides are
parallel.
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The Parallelogram 381
D C Quadrilateral ABCD is a parallelogram because AB (cid:3) CD and BC (cid:3) DA.
The symbol for parallelogram ABCDis ~ABCD.
Note the use of arrowheads, pointing in the same direction, to show sides
A B that are parallel in the figure.
Theorem 10.1 A diagonal divides a parallelogram into two congruent triangles.
Given Parallelogram ABCDwith diagonal AC D C
Prove (cid:3)ABC(cid:2)(cid:3)CDA
A B
Proof Since opposite sides of a parallelogram are parallel,alternate interior angles
can be proved congruent using the diagonal as the transversal.
Statements Reasons
1. ABCDis a parallelogram. 1.Given.
2. AB (cid:3) CDand BC (cid:3) DA 2.A parallelogram is a quadrilateral
in which two pairs of opposite
sides are parallel.
3. (cid:2)BAC(cid:2)(cid:2)DCAand 3.If two parallel lines are cut by a
(cid:2)BCA(cid:2)(cid:2)DAC transversal,alternate interior
angles are congruent.
4. AC > AC 4.Reflexive property of congruence.
5. (cid:3)ABC(cid:2)(cid:3)CDA 5.ASA.
We have proved that the diagonal AC divides parallelogram ABCD into
two congruent triangles. An identical proof could be used to show that BD
divides the parallelogram into two congruent triangles,(cid:3)ABD(cid:2)(cid:3)CDB.
The following corollaries result from this theorem.
Corollary 10.1a Opposite sides of a parallelogram are congruent.
Corollary 10.1b Opposite angles of a parallelogram are congruent.
The proofs of these corollaries are left to the student. (See exercises 14
and15.)
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382 Quadrilaterals
We can think of each side of a parallelogram as a segment of a transver-
salthat intersects a pair of parallel lines.This enables us to prove the following
theorem.
Theorem 10.2 Two consecutive angles of a parallelogram are supplementary.
Proof In ~ABCD,opposite sides are parallel.If
two parallel lines are cut by a transversal, D C
then two interior angles on the same side
of the transversal are supplementary.
Therefore,(cid:2)Ais supplementary to (cid:2)B,
(cid:2)Bis supplementary to (cid:2)C,(cid:2)Cis supple-
A B
mentary to (cid:2)D,and (cid:2)Dis supplementary
to (cid:2)A.
Theorem 10.3 The diagonals of a parallelogram bisect each other.
Given ~ABCDwith diagonals ACand BDintersecting at E. D C
Prove ACand BDbisect each other. E
A B
Proof Statements Reasons
1. AB (cid:3) CD 1.Opposite sides of a parallelogram
are parallel.
D C 2. (cid:2)BAE (cid:2)(cid:2)DCE and 2.If two parallel lines are cut by a
(cid:2)ABE (cid:2)(cid:2)CDE transversal,the alternate interior
angles are congruent.
E
A B 3. AB > CD 3.Opposite sides of a parallelogram
are congruent.
D C 4. (cid:3)ABE (cid:2)(cid:3)CDE 4.ASA.
5. AE > CEand BE > DE 5.Corresponding part of congruent
E triangles are congruent.
A B
6. Eis the midpoint of ACand 6.The midpoint of a line segment
of BD. divides the segment into two con-
gruent segments.
7. ACand BDbisect each other. 7.The bisector of a line segment
intersects the segment at its mid-
point.
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The Parallelogram 383
DEFINITION
The distance between two parallel lines is the length of the perpendicular from
any point on one line to the other line.
Properties of a Parallelogram
1. Opposite sides are parallel.
2. A diagonal divides a parallelogram into two congruent triangles.
3. Opposite sides are congruent.
4. Opposite angles are congruent.
5. Consecutive angles are supplementary.
6. The diagonals bisect each other.
EXAMPLE 1
In ~ABCD,m(cid:2)Bexceeds m(cid:2)Aby 46 degrees.Find m(cid:2)B.
Solution Let x(cid:3)m(cid:2)A.
Then x(cid:2)46 (cid:3)m(cid:2)B.
Two consecutive angles of a parallelogram are supplementary.Therefore,
m(cid:2)A(cid:2)m(cid:2)B(cid:3)180 m(cid:2)B(cid:3)x(cid:2)46
x (cid:2)x(cid:2)46 (cid:3)180 (cid:3)67(cid:2)46
2x(cid:2)46 (cid:3)180 (cid:3)113
2x(cid:3)134
x(cid:3)67
Answer m(cid:2)B (cid:3)113
Exercises
Writing About Mathematics
1. Theorem 10.2 states that two consecutive angles of a parallelogram are supplementary.If
two opposite angles of a quadrilateral are supplementary,is the quadrilateral a parallelo-
gram? Justify your answer.
2. A diagonal divides a parallelogram into two congruent triangles.Do two diagonals divide a
parallelogram into four congruent triangles? Justify your answer.
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384 Quadrilaterals
Developing Skills
3. Find the degree measures of the other three angles of a parallelogram if one angle measures:
a.70 b.65 c.90 d.130 e.155 f.168
In 4–11,ABCDis a parallelogram.
D C
4. The degree measure of (cid:2)Ais represented by 2x(cid:4)20 and the degree
measure of (cid:2)Bby 2x.Find the value of x,of m(cid:2)A,and of m(cid:2)B.
5. The degree measure of (cid:2)Ais represented by 2x(cid:2)10 and the degree
A B
measure of (cid:2)Bby 3x.Find the value of x,of m(cid:2)A,and of m(cid:2)B.
6. The measure of (cid:2)Ais 30 degrees less than twice the measure of (cid:2)B.Find the measure of
each angle of the parallelogram.
7. The measure of (cid:2)Ais represented by x (cid:2)44 and the measure of (cid:2)Cby 3x.Find the mea-
sure of each angle of the parallelogram.
8. The measure of (cid:2)B is represented by 7xand m(cid:2)Dby 5x(cid:2)30.Find the measure of each
angle of the parallelogram.
9. The measure of (cid:2)Cis one-half the measure of (cid:2)B.Find the measure of each angle of the
parallelogram.
10. If AB(cid:3)4x(cid:2)7 and CD(cid:3)3x(cid:2)12,find AB andCD.
11. If AB(cid:3)4x (cid:2)y,BC(cid:3)y(cid:2)4,CD(cid:3)3x(cid:2)6,and DA(cid:3)2x (cid:2)y,find the lengths of the sides
of the parallelogram.
12. The diagonals of ~ABCDintersect at E.If AE(cid:3)5x(cid:4)3 and EC(cid:3)15 (cid:4)x,find AC.
13. The diagonals of~ABCDintersect at E.If DE(cid:3)4y(cid:2)1 and EB(cid:3)5y(cid:4)1,find DB.
Applying Skills
14. Prove Corollary 10.1a,“Opposite sides of a parallelogram are congruent.”
15. Prove Corollary 10.1b,“Opposite angles of a parallelogram are congruent.”
16. Given:Parallelogram EBFDand parallelogram ABCDwith D C F
EABand DCF
Prove:(cid:3)EAD(cid:2)(cid:3)FCB
E A B
17. Petrina said that the floor of her bedroom is in the shape of a parallelogram and that at
least one of the angles is a right angle.Show that the floor of Petrina’s bedroom has four
right angles.
18. The deck that Jeremiah is building is in the shape of a quadrilateral,ABCD.The measure of
the angle at Ais not equal to the measure of the angle at C.Prove that the deck is not in
the shape of a parallelogram.
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Proving That a Quadrilateral Is a Parallelogram 385
19. Quadrilaterals ABCDand PQRSare parallelograms with AB > PQand BC > QR.Prove
that ABCD (cid:2)PQRSor draw a counterexample to show that they may not be congruent.
20. Quadrilaterals ABCDand PQRSare parallelograms with AB > PQ,BC > QR,and
(cid:2)B (cid:2)(cid:2)Q.Prove that ABCD (cid:2)PQRSor draw a counterexample to show that they may
not be congruent.
10-3 PROVING THAT A QUADRILATERAL IS A PARALLELOGRAM
If we wish to prove that a certain quadrilateral is a parallelogram,we can do so
by proving its opposite sides are parallel,thus satisfying the definition of a par-
allelogram.Now we want to determine other ways of proving that a quadrilat-
eral is a parallelogram.
Theorem 10.4 If both pairs of opposite sides of a quadrilateral are congruent,the quadri-
lateral is a parallelogram.
Given Quadrilateral ABCDwith AB > CD,AD > BC D C
Prove ABCDis a parallelogram.
Proof In ABCD,ACis a diagonal.Triangles ABCand CDA A B
are congruent by SSS.Corresponding parts of congru-
ent triangles are congruent,so (cid:2)BAC (cid:2)(cid:2)DCAand (cid:2)DAC (cid:2)(cid:2)ACB.ACis a
transversal that cuts ABand DC.Alternate interior angles (cid:2)BACand (cid:2)DCA
are congruent,so AB (cid:3) DC.ACis also a transversal that cuts ADand BC.
Alternate interior angles (cid:2)DAC and (cid:2)ACB are congruent so AD (cid:3) BC.
Therefore,ABCDis a parallelogram.
Theorem 10.5 If one pair of opposite sides of a quadrilateral is both congruent and
parallel,the quadrilateral is a parallelogram.
Given Quadrilateral ABCD with AB (cid:3) CDand AB > CD D C
Prove ABCDis a parallelogram.
Proof Since ABis parallel to CD,(cid:2)BACand (cid:2)DCAare a A B
pair of congruent alternate interior angles.Therefore,by
SAS,(cid:3)DCA(cid:2)(cid:3)BAC.Corresponding parts of congruent triangles are con-
gruent,so (cid:2)DAC(cid:2)(cid:2)ACB.ACis a transversal that cuts ADand BC,forming
congruent alternate interior angles (cid:2)DAC and (cid:2)ACB.Therefore,AD (cid:3) BC,
and ABCDis a parallelogram.
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Theorem 10.6 If both pairs of opposite angles of a quadrilateral are congruent,the quadri-
lateral is a parallelogram.
Given Quadrilateral ABCD with (cid:2)A(cid:2)(cid:2)Cand (cid:2)B(cid:2)(cid:2)D D C
Prove ABCDis a parallelogram.
A B
Proof The sum of the measures of the angles of a quadrilateral
is 360 degrees.Therefore,m(cid:2)A(cid:2)m(cid:2)B(cid:2)m(cid:2)C(cid:2)m(cid:2)D(cid:3)360.It is given
that (cid:2)A(cid:2)(cid:2)Cand (cid:2)B(cid:2)(cid:2)D.Congruent angles have equal measures so
m(cid:2)A (cid:3)m(cid:2)C and m(cid:2)B(cid:3)m(cid:2)D.
By substitution,m(cid:2)A(cid:2)m(cid:2)D(cid:2)m(cid:2)A(cid:2)m(cid:2)D(cid:3)360.Then,
2m(cid:2)A(cid:2)2m(cid:2)D(cid:3)360 or m(cid:2)A(cid:2)m(cid:2)D(cid:3)180.Similarly,m(cid:2)A (cid:2)m(cid:2)B(cid:3)180.
If the sum of the measures of two angles is 180,the angles are supplemen-
tary.Therefore,(cid:2)Aand (cid:2)Dare supplementary and (cid:2)A and (cid:2)Bare
supplementary.
Two coplanar lines are parallel if a pair of interior angles on the same side
of the transversal are supplementary.Therefore,AB (cid:3) DCand AD (cid:3) BC.
Quadrilateral ABCD is a parallelogram because it has two pairs of parallel
sides.
Theorem 10.7 If the diagonals of a quadrilateral bisect each other, the quadrilateral is a
parallelogram.
Given Quadrilateral ABCD with ACand BDintersecting at E, D C
AE > EC,BE > ED
E
Prove ABCDis a parallelogram.
A B
Strategy Prove that (cid:3)ABE(cid:2)(cid:3)CDEto show that one pair of
opposite sides is congruent and parallel.
The proof of Theorem 10.7 is left to the student.(See exercise 15.)
SUMMARY To prove that a quadrilateral is a parallelogram, prove that any
one of the following statements is true:
1. Both pairs of opposite sides are parallel.
2. Both pairs of opposite sides are congruent.
3. One pair of opposite sides is both congruent and parallel.
4. Both pairs of opposite angles are congruent.
5. The diagonals bisect each other.
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Proving That a Quadrilateral Is a Parallelogram 387
EXAMPLE 1
Given:ABCDis a parallelogram.
D F C
Eis on AB,Fis on DC,and EB > DF.
Prove:DE (cid:3) FB
A E B
Proof We will prove that EBFDis a parallelogram.
Statements Reasons
1. ABCDis a parallelogram. 1.Given.
2. AB (cid:3) DC 2.Opposite sides of a parallelogram
are parallel.
3. EB (cid:3) DF 3.Segments of parallel lines are
parallel.
4. EB > DF 4.Given.
5. EBFD is a parallelogram. 5.If one pair of opposite sides of a
quadrilateral is both congruent
and parallel,the quadrilateral is a
parallelogram.
6. DE (cid:3) FB 6.Opposite sides of a parallelogram
are parallel.
Exercises
Writing About Mathematics
1. What statement and reason can be added to the proof in Example 1 to prove that DE > FB?
2. What statement and reason can be added to the proof in Example 1 to prove that
(cid:2)DEB (cid:2)(cid:2)BFD?
Developing Skills
In 3–7, in each case, the given is marked on the figure. Tell why each quadrilateral ABCD is a
parallelogram.
3. D C 4. D C 5. D C 6. D C 7. D C
E
A B A B A B A B A B
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388 Quadrilaterals
8. ABCDis a quadrilateral with AB (cid:3) CDand (cid:2)A(cid:2)(cid:2)C.Prove that ABCDis a parallelo-
gram.
9. PQRSis a quadrilateral with (cid:2)P(cid:2)(cid:2)R and (cid:2)P the supplement of (cid:2)Q.Prove that PQRSis
a parallelogram.
10. DEFGis a quadrilateral with DFdrawn so that (cid:2)FDE (cid:2)(cid:2)DFGand (cid:2)GDF (cid:2)(cid:2)EFD.
Prove thatDEFGis a parallelogram.
11. ABCDis a parallelogram.Eis the midpoint of ABand Fis the midpoint of CD.Prove that
AEFD is a parallelogram.
h
12. EFGH is a parallelogram and Jis a point on EFsuch that Fis the midpoint of EJ.Prove
that FJGHis a parallelogram.
13. ABCD is a parallelogram.The midpoint of ABis P,the midpoint of BCis Q,the midpoint
of CDis R,and the midpoint of DAis S.
a. Prove that (cid:3)APS(cid:2)(cid:3)CRQand that (cid:3)BQP(cid:2)(cid:3)DSR.
b. Prove that PQRSis a parallelogram.
14. A quadrilateral has three right angles.Is the quadrilateral a parallelogram? Justify your
answer.
Applying Skills
15. Prove Theorem 10.7,“If the diagonals of a quadrilateral bisect each other,the quadrilateral
is a parallelogram.”
16. Prove that a parallelogram can be drawn by joining the endpoints of two line segments that
bisect each other.
17. The vertices of quadrilateral ABCDare A((cid:4)2,1),B(4,(cid:4)2),C(8,2),and D(2,5).Is ABCD a
parallelogram? Justify your answer.
18. Farmer Brown’s pasture is in the shape of a quadrilateral,PQRS.The pasture is crossed by
two diagonal paths,PRand QS.The quadrilateral is not a parallelogram.Show that the
paths do not bisect each other.
19. Toni cut two congruent scalene triangles out of cardboard.She labeled one triangle (cid:3)ABC
and the other (cid:3)A(cid:5)B(cid:5)C(cid:5)so that (cid:3)ABC (cid:2)(cid:3)A(cid:5)B(cid:5)C(cid:5).She placed the two triangles next to
each other so that Acoincided with B(cid:5)and Bcoincided with A(cid:5).Was the resulting quadrilat-
eral ACBC(cid:5)a parallelogram? Prove your answer.
20. Quadrilateral ABCDis a parallelogram with Mthe midpoint of ABand N the midpoint of
CD.
a. Prove that AMNDand MBCNare parallelograms.
b. Prove that AMNDand MBCN are congruent.